An electric dipole with dipole moment $p$ is in a uniform external electric field $E$ . Show that for the stable orientation in part (b), the dipole's own electric field tends to oppose the external field. Note: Part (b) asked which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small rotation away from the equilibrium position and see what happens.) Also, part (a) asked to find the orientations of the dipole for which the torque on the dipole is zero.

검증된 단계별 안내

Understand the concept of an electric dipole: An electric dipole consists of two equal and opposite charges separated by a distance. The dipole moment $ \mathbf{p} $ is a vector quantity defined as $ \mathbf{p} = q \cdot \mathbf{d} $, where $ q $ is the charge and $ \mathbf{d} $ is the displacement vector from the negative to the positive charge.

Recall the torque on a dipole in an electric field: The torque $ \mathbf{\tau} $ experienced by a dipole in a uniform electric field $ \mathbf{E} $ is given by $ \mathbf{\tau} = \mathbf{p} \times \mathbf{E} $. This torque tends to align the dipole with the electric field.

Identify the stable orientation: The stable orientation of a dipole in an electric field is when the dipole moment $ \mathbf{p} $ is aligned with the electric field $ \mathbf{E} $. In this orientation, the angle $ \theta $ between $ \mathbf{p} $ and $ \mathbf{E} $ is zero, minimizing the potential energy.

Consider the dipole's own electric field: The electric field due to the dipole itself is directed from the positive to the negative charge. In the stable orientation, this field is opposite to the direction of the external electric field $ \mathbf{E} $.

Conclude the opposition of fields: Since the dipole's own electric field is directed opposite to the external field in the stable orientation, it tends to oppose the external field, thereby stabilizing the dipole in that orientation.

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주요 개념

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Electric Dipole Moment

The electric dipole moment is a vector quantity that represents the separation of positive and negative charges within a system. It is defined as the product of the charge magnitude and the distance between the charges, pointing from the negative to the positive charge. In an external electric field, the dipole moment determines the torque experienced by the dipole.

Torque on a Dipole

Torque on an electric dipole in an external electric field is given by the cross product of the dipole moment and the electric field vector. This torque tends to align the dipole moment with the electric field, resulting in a stable orientation when the dipole moment is parallel to the field. The torque is zero when the dipole is aligned, indicating a stable equilibrium.

Electric Field of a Dipole

The electric field generated by a dipole is a vector field that varies with position around the dipole. It tends to oppose external fields when the dipole is in a stable orientation, as the dipole's field lines are directed opposite to the external field lines. This opposition is due to the dipole's attempt to minimize potential energy by aligning with the external field.

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교과서 질문

The dipole moment of the water molecule (H₂O) is $6.17 \times 10^{- 30}$ Cm. Consider a water molecule located at the origin whose dipole moment $p$ points in the $+ x$ -direction. A chlorine ion (Cl^-), of charge $- 1.60 \times 10^{- 19}$ C, is located at $x = 3.00 \times 10^{- 9}$ m. Find the magnitude and direction of the electric force that the water molecule exerts on the chlorine ion. Is this force attractive or repulsive? Assume that $x$ is much larger than the separation $d$ between the charges in the dipole, so that the approximate expression for the electric field along the dipole axis derived in Example $21.14$ can be used.

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교과서 질문

Point charges $q_{1} = - 4.5$ nC and $q sub 2 equals positive 4.5$ nC are separated by $3.1$ mm, forming an electric dipole. The charges are in a uniform electric field whose direction makes an angle of $36.9$ ° with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude $7.2 \times 10^{- 9}$ Nm?

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$ . Find the orientations of the dipole for which the torque on the dipole is zero.

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교과서 질문

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$ . Which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small rotation away from the equilibrium position and see what happens.) Note: Part (a) asked to find the orientations of the dipole for which the torque on the dipole is zero.

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